Best approach for 2D Grid Image Segmentation
This is a really cool problem. You already have a working model here are a few different ways of going forward with the project.


Grouping text based on locality. "no segmentation"  Text region extraction in a document image based on the Delaunay tessellation 
Segmentation  Multiscale Edge-Based Text Extraction from Complex Images 
Training a map of the image then extracting text form the map.  Multi Oriented Text Detection 


Training the model to detect boxes followed by extracting the text from each box seems like a very smart direction to move in. This paper talks about the former.  automatic image segmentation and edge detection  Good luck!